bayesian model selection for spatial sound field reconstruction: a comparative analysis of empirical and nested sampling methods

Abstract

The Plane Wave Decomposition model provides a robust framework to simplify and recreate a complex sound system-under-test in a localized area. However, model selection and parameter estimation in acoustical signal reconstruction face persistent challenges due to ill-posed inverse algebraic problems, model complexity, and probabilistic interdependence of parameters. Goals surrounding realistic applications of this analysis problem often involve finding an equilibrium between auralization accuracy and the cardinality of the model. This research presents a comprehensive comparison between analytical Empirical Bayes (EB) methods and stochastic Nested Sampling (NS) algorithms. Within the EB framework, a novel stabilization routine using Truncated Singular Value Decomposition (SVD) is implemented to overcome the numerical failures of standard numerical estimation methods in high-dimensional parameter spaces. To provide a more rigorous estimation of the marginal likelihood, a "Guided" Nested Sampling strategy is developed. By utilizing a Maximum A Posteriori (MAP) seed to anchor the initial live point set, the algorithm successfully navigates high-dimensional ridges that are otherwise unreachable via blind uniform sampling. The results demonstrate that as model cardinality and frequency increase, the Nested Sampling evidence provides a more accurate representation of the sound field over (EB), and successfully captures the details within the iso-likelihood of the parameter space, which distribution approximations otherwise flatten. This research further advocates that the nested sampling routine is well-suited to estimate the optimal cardinality of a plane wave decomposition basis set. {\bf\noindent Key Words:} Plane Wave Decomposition; Wave Field Synthesis; Bayesian Analysis; Nested Sampling; Model Selection; Acoustics